The flight crew must be certified to perform higher category approaches. And, yes, Cat III approaches have MUCH lower mins (right down to 0-0) than a Cat I. No, it doesn't stop at III... not really. It goes:
with CAT IIIc having zero-zero landing mins, and a requirement for specialized flight crew, instruments, and manditory autoland.
If you do a forum search, you can find a LOT more detailed explanation.